## X(173) (CONGRUENT ISOSCELIZERS POINT)

 Interactive Applet

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 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos B/2 + cos C/2 - cos A/2
Trilinears           tan(A/2) + sec(A/2) : tan(B/2) + sec(B/2) : tan(C/2) + sec(C/2)      (M. Iliev, 4/12/07)

Barycentrics    g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

Let P(B)Q(C) be an isoscelizer: let P(B) on sideline AC and Q(C) on AB be equidistant from A, so that AP(B)Q(C) is an isosceles triangle. Line P(B)-to-Q(C), P(C)-to-Q(A), P(A)-to-Q(B) concur in X(173). (P. Yff, unpublished notes, 1989)

The intouch triangle of the intouch triangle of triangle ABC is perspective to triangle ABC, and X(173) is the perspector. (Eric Danneels, Hyacinthos 7892, 9/13/03)

Also, X(173) = X(1486)-of-the-intouch-triangle. (Darij Grinberg; see notes at X(1485) and X(1486).)

Congruent Isoscelizers Point.

X(173) lies on these lines: 1,168    9,177    57,174    164,504    180,483    503,844    505,1130

X(173) = isogonal conjugate of X(258)
X(173) = X(174)-Ceva conjugate of X(1)

X(173) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,503), (2,504), (174,173), (188,845), (366,164), (507,1), (508,362), (509,361)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.